On this page you’ll learn an introduction to two essential trigonometric functions, the sine and cosine functions. These functions relate the angles and sides of a right triangle and have numerous applications in fields such as engineering, physics, and astronomy. The page includes the definitions, formulas, and tables of values for these functions, making it a valuable resource for anyone studying or working in technical fields.
Sine Theta (sin θ)
The sine function is a trigonometric function that relates the angle θ to the ratio of the length of the side opposite to the angle to the length of the hypotenuse of a right triangle.
- Formula: sin θ = opposite/hypotenuse
- Unit circle: The sine function can also be represented on the unit circle, which is a circle with a radius of 1 unit, where the angle θ is measured from the positive x-axis in the counterclockwise direction. The sine function of an angle θ is equal to the y-coordinate of the point on the unit circle that corresponds to the angle θ.
- Periodicity: The sine function is periodic, with a period of 2π. This means that sin(θ + 2π) = sin(θ) for any angle θ.
- Graph: The graph of the sine function is a periodic wave that oscillates between -1 and 1, with a wavelength of 2π.
| θ | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 | -1 | 0 |
Cosine Theta (cos θ)
The cosine function is a trigonometric function that relates the angle θ to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse of a right triangle.
- Formula: cos θ = adjacent/hypotenuse
- Unit circle: The cosine function can also be represented on the unit circle. The cosine function of an angle θ is equal to the x-coordinate of the point on the unit circle that corresponds to the angle θ.
- Periodicity: The cosine function is periodic, with a period of 2π. This means that cos(θ + 2π) = cos(θ) for any angle θ.
- Graph: The graph of the cosine function is also a periodic wave that oscillates between -1 and 1, with a wavelength of 2π.
| θ | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 | -1 | 0 | 1 |
These tables can be useful when solving problems that involve trigonometric functions, such as finding the length of a side of a right triangle or the value of an angle given the lengths of two sides.